I think it depends on what kind of fermenter you are using. For example, if your fermenter is plastic (PET) like a fermzilla or better bottle I wouldn't leave it in there for extended periods of time as PET's oxygen barrier properties aren't suitable for long term aging.
I also wouldn't recommend leaving it in a glass carboy where you can't maintain positive pressure. Oxygen will also creep in and ruin your beer. You also need to be very careful cold crashing in a carboy that can't hold pressure as air will be sucked into the fermentor from the drop in temperature (vacuum is formed)
However, if your primary fermenter is a conical or keg that can maintain pressure (and made from stainless steel) there's no reason you couldn't do it in the primary vessel (ideally after pulling out the yeast)
If you keg beer, the simplest thing to do is just lager in the serving keg (using a floating diptube). Essentially just lager by packaging it and sticking it in your keezer on gas
If you don't have pressure capable primary fermenters and are a bottler, then I would just recommend bottle lagering. Basically package your beer as normal, carbonate in the bottles, then let the bottles sit in the fridge for a few weeks after carbonated. Be careful not to disturb the yeast sediment when pouring